It is shown that the existence of a parallel (-)-spinor with respect to a metric connection with totally skew-symmetric torsion requires more local restrictions than the existence of a parallel (+)-spinor. It is proved that every harmonic spinor with respect to the Dirac operator of this connection on a compact four-dimensional spin Riemannian manifold is parallel with respect to a naturally arising metric connection with totally skew-symmetric torsion and all such spaces are classified up to a conformal transformation.

have shown that the infinitesimal period map for an algebraic completely integrable Hamiltonian system (ACIHS) is encoded in a section of the third symmetric power of the cotangent bundle to the base of the system. For the ordinary Hitchin system the cubic is given by a formula of Balduzzi and Pantev. We show that the Balduzzi-Pantev formula holds on maximal rank symplectic leaves of the G-generalised Hitchin system.

Abstract. The present note is mostly a survey on the generalised Hitchin integrable system and moduli spaces of meromorphic Higgs bundles. We also fill minor gaps in the existing literature, outline a calculation of the infinitesimal period map and review briefly some related geometries.